Idea

In principle also all other notions of theory, such as in the sense of physics should be special cases of this, but in practice of course there are many systems called “theories” which are not (yet) as fully formalized as in mathematical logic.

Definition

is that the theory itself consists of a set of formulas in the first order language Lang(Σ)Lang(\Sigma) of a signatureΣ\Sigma. Classically, these formulas are assumed to have no free variables (i.e. to be “sentences”), but in weaker logics that lack universal quantification it is better to take them to be formulas-in-context. One also sometimes considers the theory to also include all logical consequences (aka theorems) of the axioms in AA, relative to (some specified) fragment of first-order logic — that is, to be “saturated” with respect to provability.

is that the theory is given by the class of its models appropriate to that fragment of logic. Gödel’s completeness theorem is that a sentence in 𝒯\mathcal{T} is a theorem iff it is satisfied in every model.

is that the logical formalism of a theory 𝒯\mathcal{T} can frequently be embodied in a syntactic category of terms C𝒯C_{\mathcal{T}}, so that models of a theory 𝒯\mathcal{T} are identified with functors

C𝒯→SetC_{\mathcal{T}} \to Set

that preserve some (typically property-like) structures on C𝒯C_{\mathcal{T}}, such as certain classes of colimits or of limits, pertinent to the fragment of logic at hand. Then a completeness theorem would be the statement that the canonical map

C𝒯→∏modelsFinSetSetC_{\mathcal{T}} \to \prod_{models F in Set} Set

is a full faithful embedding (one that preserves all relevant logical structure). For this reason, completeness theorems are also known as embedding theorems.

Hm, is that the way it should be said?

In fact, the notion of model can be generalized away from SetSet to more general categories, namely those that have enough structure to “internalize” the fragment of logic at hand. From this very general point of view on model, the syntactic category C𝒯C_{\mathcal{T}} is the generic or universal model for 𝒯\mathcal{T}, and if we simply call C𝒯C_{\mathcal{T}} the theory, then models and theories are placed on the same footing.

Hierarchy of theories: cartesian, regular, coherent, geometric

There is a hierarchy of theories that can be interpreted in the internal logic of a hierarchy of types of categories. Since predicates in the internal logic are represented by subobjects, in order to interpret any connective or quantifier in the internal logic, one needs a corresponding operation on subobjects to exist in the category in question, and be well-behaved. For instance:

Note that the axioms of one of these theories are actually of the form

φ⊢x⇀ψ \varphi \vdash_{\vec{x}} \psi

where φ\varphi and ψ\psi are formulas involving only the specified connectives and quantifiers, ⊢\vdash means entailment, and x⇀\vec{x} is a context. Such an axiom can also be written as

∀x⇀.(φ⇒ψ) \forall \vec{x}. (\varphi \Rightarrow \psi)

so that although ⇒\Rightarrow and ∀\forall are not strictly part of any of the above logics, they can be applied “once at top level.” In an axiom of this form for geometric logic, the formulas φ\varphi and ψ\psi which must be built out of ⊤\top, ∧\wedge, ⊥\bot, ⋁\bigvee, and ∃\exists are sometimes called positive formulas.

Abelian theories

Interestingly, one form of logic which made an early appearance but is not ordinarily thought of as logic at all is the logic of abelian categories, which is characterized by certain exactness properties. Here a small abelian category AA can be thought of as a syntactic site for some “abelian theory”; models of the theory are exact additive functors with domain AA. The classical models would in fact be exact additive functors A→AbA \to Ab, or exact additive functors to a category of modules. A “Freyd-Heron-Lubkin-Mitchell” embedding theorem is then a completeness theorem with respect to the classical models, and assures us that a statement in the language of abelian category theory is provable if and only if it is true when interpreted in any module category.

Models for a theory

Set-theoretic models for a first-order theory in syntactic approach

The basic concept is of a structure for a first-order language LL: a set MM together with an interpretation of LL in MM. A theory is specified by a language and a set of sentences in LL. An LL-structure MM is a model of TT if for every sentence ϕ\phi in TT, its interpretation in MM, ϕM\phi^M is true (“ϕ\phi holds in MM”). We say that TT is consistent or satisfiable (relative to the universe in which we do model theory) if there exist at least one model for TT (in our universe). Two theories, T1T_1, T2T_2 are said to be equivalent if they have the same models.

Given a class KK of structures for LL, there is a theory Th(K)Th(K) consisting of all sentences in LL which hold in every structure from KK. Two structures MM and NN are elementary equivalent (sometimes written by equality M=NM=N, sometimes said “elementarily equivalent”) if Th(M)=Th(N)Th(M)=Th(N), i.e. if they satisfy the same sentences in LL. Any set of sentences which is equivalent to Th(K)Th(K) is called a set of axioms of KK. A theory is said to be finitely axiomatizable if there exist a finite set of axioms for KK.

A theory is said to be complete if it is equivalent to Th(M)Th(M) for some structure MM.

Categorical point of view and models in topoi

From the categorical point of view, for every theory TT there exists a categoryCTC_T – the syntactic categoryCTC_T – such that a model for TT is a functorCT→𝒯C_T \to \mathcal{T} into some toposTT, satisfying certain conditions.

For instance the syntactic categories of Lawvere theories are precisely those categories that have finite cartesian products and in which every object is isomorphic to a finite cartesian power xnx^n of a distinguished object xx. A model for a Lawvere theory is precisely a finite product preserving functor CT→𝒯C_T \to \mathcal{T}.

We say a functor 𝒯1→𝒯2\mathcal{T}_1 \to \mathcal{T}_2 of toposes (for instance a logical morphism or a geometric morphism) preserves a theoryTT if for every model CT→𝒯1C_T \to \mathcal{T}_1 of TT in 𝒯1\mathcal{T}_1, the composite CT→𝒯1→𝒯2C_T \to \mathcal{T}_1 \to \mathcal{T}_2 is a model of TT in 𝒯2\mathcal{T}_2.